Theorem curry1val 8109

Description: The value of a curried function with a constant first argument. (Contributed by NM, 28-Mar-2008.) (Revised by Mario Carneiro, 26-Apr-2015.)

Hypothesis

Ref	Expression
curry1.1	⊢ 𝐺 = (𝐹 ∘ ◡(2nd ↾ ({𝐶} × V)))

Assertion

Ref	Expression
curry1val	⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → (𝐺‘𝐷) = (𝐶𝐹𝐷))

Proof of Theorem curry1val

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		curry1.1	. . . 4 ⊢ 𝐺 = (𝐹 ∘ ◡(2nd ↾ ({𝐶} × V)))
2	1	curry1 8108	. . 3 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → 𝐺 = (𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥)))
3	2	fveq1d 6883	. 2 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → (𝐺‘𝐷) = ((𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥))‘𝐷))
4		eqid 2736	. . . . . . 7 ⊢ (𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥)) = (𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥))
5	4	fvmptndm 7022	. . . . . 6 ⊢ (¬ 𝐷 ∈ 𝐵 → ((𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥))‘𝐷) = ∅)
6	5	adantl 481	. . . . 5 ⊢ (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) ∧ ¬ 𝐷 ∈ 𝐵) → ((𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥))‘𝐷) = ∅)
7		fndm 6646	. . . . . . 7 ⊢ (𝐹 Fn (𝐴 × 𝐵) → dom 𝐹 = (𝐴 × 𝐵))
8	7	adantr 480	. . . . . 6 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → dom 𝐹 = (𝐴 × 𝐵))
9		simpr 484	. . . . . . 7 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → 𝐷 ∈ 𝐵)
10	9	con3i 154	. . . . . 6 ⊢ (¬ 𝐷 ∈ 𝐵 → ¬ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵))
11		ndmovg 7595	. . . . . 6 ⊢ ((dom 𝐹 = (𝐴 × 𝐵) ∧ ¬ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵)) → (𝐶𝐹𝐷) = ∅)
12	8, 10, 11	syl2an 596	. . . . 5 ⊢ (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) ∧ ¬ 𝐷 ∈ 𝐵) → (𝐶𝐹𝐷) = ∅)
13	6, 12	eqtr4d 2774	. . . 4 ⊢ (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) ∧ ¬ 𝐷 ∈ 𝐵) → ((𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥))‘𝐷) = (𝐶𝐹𝐷))
14	13	ex 412	. . 3 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → (¬ 𝐷 ∈ 𝐵 → ((𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥))‘𝐷) = (𝐶𝐹𝐷)))
15		oveq2 7418	. . . 4 ⊢ (𝑥 = 𝐷 → (𝐶𝐹𝑥) = (𝐶𝐹𝐷))
16		ovex 7443	. . . 4 ⊢ (𝐶𝐹𝐷) ∈ V
17	15, 4, 16	fvmpt 6991	. . 3 ⊢ (𝐷 ∈ 𝐵 → ((𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥))‘𝐷) = (𝐶𝐹𝐷))
18	14, 17	pm2.61d2 181	. 2 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → ((𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥))‘𝐷) = (𝐶𝐹𝐷))
19	3, 18	eqtrd 2771	1 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → (𝐺‘𝐷) = (𝐶𝐹𝐷))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3464  ∅c0 4313  {csn 4606   ↦ cmpt 5206   × cxp 5657  ^◡ccnv 5658  dom cdm 5659   ↾ cres 5661   ∘ ccom 5663   Fn wfn 6531  ‘cfv 6536  (class class class)co 7410  2^nd c2nd 7992

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-1st 7993  df-2nd 7994

This theorem is referenced by:  nvinvfval  30626  hhssabloilem  31247